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Single-phase center-tapped (or split-phase) transformer has a single-phase input winding and a center-tapped output winding connected to a grounded neutral as seen in the figure below. This model is developed from a three winding single-phase transformer whose polarity of the tertiary winding respect to ground has been inverted to obtain twice the phase-ground voltage between secondary and tertiary.
Single-phase 3W Transformer
Symbol | Description | Unit | |
| ID | Transformer name | unique name | |
|---|---|---|---|
Status | Connect/Disconnect status | Initial value 1 (0 for disconnected) | |
| Number of Phases | Phase count in use | for this model only 1 phase is accepted | |
Primary winding From | Bus1 | Primary side: Bus 1 | a unique name |
Bus2 | Primary side: Bus 2 | NOT APPLICABLE | |
Bus3 | Primary side: Bus 3 | NOT APPLICABLE | |
V (kV) | Primary winding nominal voltage (phase-to-ground) | kV | |
S_base (kVA) | Nominal power in primary side | kVA | |
Conn. type (*) | Primary winding connection type | for this model only ‘wye’ connection is accepted | |
Secondary winding To | Bus1 | Secondary side: Bus 1 | a unique name |
Bus2 | Secondary side: Bus 2 | NOT APPLICABLE | |
Bus3 | Secondary side: Bus 3 | NOT APPLICABLE | |
V (kV) | Secondary winding nominal voltage (phase-to-ground) | kV | |
S_base (kVA) | Nominal power in secondary side | NOT APPLICABLE | |
Conn. type (*) | Secondary winding connection type | for this model only ‘wye’ connection is accepted | |
Tertiary winding To | Bus1 | Tertiary side: Bus 1 | a unique name |
Bus2 | Tertiary side: Bus 2 | NOT APPLICABLE | |
Bus3 | Tertiary side: Bus 3 | NOT APPLICABLE | |
V (kV) | Tertiary winding nominal voltage (phase-to-ground) | kV | |
S_base (kVA) | Nominal power in tertiary side | NOT APPLICABLE | |
Conn. type (*) | Tertiary winding connection type | for this model only ‘wye’ connection is accepted | |
R_12 (p.u.) | Resistance between primary and secondary windings | p.u. | |
Xl_12 (p.u.) | Reactance between primary and secondary windings | p.u. | |
R_23 (p.u.) | Resistance between secondary and tertiary windings | p.u. | |
Xl_23 (p.u.) | Reactance between secondary and tertiary windings | p.u. | |
R_31 (p.u.) | Resistance between primary and tertiary windings | p.u | |
Xl_31 (p.u.) | Reactance between primary and tertiary windings | p.u. | |
Operation | Type of operation: 0 = normal 3W trx, 1 = center-tapped trx | 0 or 1 | |
Model Equations
This single-phase transformer is modeled based on the primitive nodal admittance matrix Yprim [1].
Yprim = CT M Yw M C matrix dimension: 3 x 3
Yw0 =A YP AT
Yw0 is the winding admittance matrix. Matrix dimension: 4 x 4
A is the admittance-winding incidence matrix whose non-zero elements are generally either 1 and -1. Matrix dimension: 4 x 4
YP is the primitive admittance matrix on a 1 V base. Matrix dimension: 4 x 4
Yw =Zw-1 matrix dimension: 3 x 3
Zw is the reduced impedance matrix. Calculated by inverting the matrix Yw0 and eliminating the row and column of the ficticious node. Matrix dimension: 3 x 3
C is the winding-port incidence matrix whose non-zero elements are generally either 1 and -1. Matrix dimension: 3 x 3
M is the incidence matrix whose non-zero elements are the inverse of the numbers of turns in the windings. Matrix dimension: 3 x 3
Example
1) A single-phase 3W transformer with the following data: 7.2/0.12/0.12 kV, 15 kVA, X12 = 1.44%, R12=1.95%, X13 = 1.44%, R13=1.95%, X23 = 0.96%, R23=2.6% will be connected like a center-tapped transformer
The matrices are calculated according the diagram shown above.
Zhl = R12 + X12i (pu) ; Zht = R13 + X13i (pu); Zlt = R23 + X23i (pu).
Calculation of Z1, Z2 and Z3. Z123=0.5 Ztrx
Ztrx=
| Zhl | -Zlt | Zht |
| Zhl | Zlt | -Zht |
| -Zhl | Zlt | Zht |
from here Z1 = 0.0065+0.0096i; Z2 = 0.013+0.0048i and Z3 = 0.013+0.0048i. Transforming to a 1V base and inverting admittances Y1, Y2 and Y3 are obtained to build matrix Yp. Assuming a high value for the impedance Z0 between ficticious node and the reference (for example 500e3 ohms)
Yw0 =A YP AT =
A =
| 1 | 0 | 0 | -1 |
| 0 | 1 | 0 | -1 |
| 0 | 0 | 1 | -1 |
| 0 | 0 | 0 | 1 |
YP =
| 7.5239e5 - 1.0713e6i | 0 | 0 | 0 |
| 0 | 1.0154e6 - 3.7492e5i | 0 | 0 |
| 0 | 0 | 1.0154e6 - 3.7492e5i | 0 |
| 0 | 0 | 0 | 0.03 |
Yw0 =
| 7.5239e5 - 1.0713e6i | 0 | 0 | -7.5239e5 + 1.0713e6i |
| 0 | 1.0154e6 - 3.7492e5i | 0 | -1.0154e6 + 3.7492e5i |
| 0 | 0 | 1.0154e6 - 3.7492e5i | -1.0154e6 + 3.7492e5i |
| -7.5239e5 + 1.0713e6i | -1.0154e6 + 3.7492e5i | -1.0154e6 + 3.7492e5i | 2.7562e6 - 1.8212e6i |
Inverting this matrix and eliminating the row and column of the ficticious node, Zw is obtained
Zw =
| 33.3333 | 33.3333 | 33.3333 |
| 33.3333 | 33.3333 | 33.3333 |
| 33.3333 | 33.3333 | 33.3333 |
and Yw =Zw-1
Yw =
| 6.23e5 - 5.7508e5i | -3.1150e5 + 2.8754e5i | -3.1150e5 + 2.8754e5i |
| -3.1150e5 + 2.8754e5i | 6.6346e5 - 3.3123e5i | -3.5196e5 + 4.3691e4i |
| -3.1150e5 + 2.8754e5i | -3.5196e5 + 4.3691e4i | 6.6346e5 - 3.3123e5i |
Matrices C and M are defined as shown below. The negative sign in the matrix C allows to change the polarity of the tertiary winding to obtain the center-tapped connection,
C =
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | -1 |
M =
| 1/7200 | 0 | 0 |
| 0 | 1/120 | 0 |
| 0 | 0 | 1/120 |
which finally allows to obtain the nodal admittance matrix of the center-tapped transformer
Yprim =
| 0.012 - 0.0111i | -0.3605 + 0.3328i | 0.3605 - 0.3328i |
| -0.3605 + 0.3328i | 46.0734 - 23.0021 | 24.4414 - 3.0341i |
| 0.3605 - 0.3328i | 24.4414 - 3.0341i | 46.0734 - 23.0021 |
To add this transformer in the excel file, see the next image
References
[1] Massimiliano Coppo, Fabio Bignucolo and Roberto Turri, "Generalised transformer modelling for power flow calculation in multi-phase unbalanced networks". IET Generation, Transmission and Distribution, 2017, Vol. 11, Issue: 15, pp: 3843-3852. DOI: 10.1049/iet-gtd.2016.2080